Camera-based speed estimation and system calibration therefor

ABSTRACT

Provided is a visual vehicle speed estimation system based on camera output and calibration for such a vehicle speed estimation system. The calibration allows use of the system where the absolute camera position is unknown. Calibration determines an absolute-position-independent relationship between an image space placement and a physical space position relative to the camera. The calibration is done on the basis of camera output using vehicle features of known dimensions and some assumed physical constraints related thereto to provide a conversion relationship between image coordinates and physical space coordinates in a physical space defined in relation to the camera. This relationship is then used to estimate vehicle speeds based only on the visual information provided by the camera. Abstract is not to be interpreted as limiting.

TECHNICAL FIELD

This invention concerns the field of vehicle speed estimation based onvisual information and the field of calibration of systems for visualvehicle speed estimation.

BACKGROUND

Provided is improved technology for vehicle speed estimation. Vehiclespeeds are estimated using visual information derived from a camera, forexample in an LPR system. Calibration is provided to allow the system toestimate vehicle speeds in a variety of setups including setups wherethe images are provided from cameras that are located off-center from atraffic lane, for example on the side of the road. Calibration may bedone with as little as a single passage of an unknown vehicle. In someembodiments it may be done alongside speed estimation with every vehicledetected/estimated.

In accordance with a non-limiting example is provided a method forcalibrating a vehicle speed estimation system that employs a camerahaving a set of camera parameters to estimate the speed of vehiclesbased on images received from the camera, the images defined in an imagespace and the vehicles and camera each belonging to a physical space andhaving an absolute position therein. The method comprises receiving at acontroller from the camera a calibration image stream comprising a firstimage of a vehicle captured at a first time and a second image of thevehicle captured at a second time, the first and second images havingthe same image space. The method further comprises at a controllerexecuting controller logic to identify in each of the first and secondimage a vehicle feature having a known dimension in the physical space,the identifying comprising determining a placement of the vehiclefeature in the image space at each of the first and second times. Themethod further comprises at the controller executing controller logic todefine on the basis of the placement of the vehicle feature in the imagespace at each of the first and second times anabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera. Themethod further comprises at the controller, using controller logic,applying the absolute-position-independent relationship between an imagespace placement and a physical space position relative to the camera ina speed estimation algorithm to determine the speed of a vehiclecaptured by the camera.

In accordance with another non-limiting example is provided acamera-based visual speed estimation system for estimating the speed ofvehicles on the basis of video output from of a camera. The camera-basedvisual speed estimation system comprises a controller camera input forreceiving from the video output of the camera comprising at least firstand second images of a travelling vehicle captured at respective firstand second times, the first and second images being defined in an imagespace and the travelling vehicle and camera each belonging to a physicalspace and having respective absolute positions therein. The camera-basedvisual speed estimation system further comprises a tangible controllerin communication with the camera input and receiving therefrom the videodata from the camera comprising at least the first and second images.The controller comprises calibration logic executable on the controllerfor causing the controller to identify in each of the first and secondimage a vehicle feature having a known dimension in the physical space,the identifying comprising determining a placement of the vehiclefeature in the image space at each of the first and second times. Thecontroller further comprises calibration logic executable on thecontroller for causing the controller to define on the basis of theplacement of the vehicle feature in the image space at each of the firstand second times an absolute-position-independent relationship betweenan image space placement and a physical space position relative to thecamera. The controller further comprises calibration logic executable onthe controller for causing the controller to apply theabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera in aspeed estimation algorithm to derive speed estimation data indicative ofthe speed of a vehicle captured by the camera. The camera-based visualspeed estimation system further comprises a controller output incommunication with the controller for receiving from the controller thespeed estimation data and outputting the speed estimation.

SUMMARY

Applicant has made a number of improvements that taken alone or incombination can provide advantages over the state-of-the-art approaches.

In accordance with a first exemplary embodiment is provided a method forcalibrating a vehicle speed estimation system that employs a camerahaving a set of camera parameters to estimate the speed of vehiclesbased on images received from the camera, the images defined in an imagespace and the vehicles and camera each belonging to a physical space andhaving an absolute position therein. The method comprises receiving at acontroller from the camera a calibration image stream comprising a firstimage of a vehicle captured at a first time and a second image of thevehicle captured at a second time, the first and second images havingthe same image space. The method further comprises at a controllerexecuting controller logic to identify in each of the first and secondimage a vehicle feature having a known dimension in the physical space,the identifying comprising determining a placement of the vehiclefeature in the image space at each of the first and second times. Themethod further comprises at the controller executing controller logic todefine on the basis of the placement of the vehicle feature in the imagespace at each of the first and second times anabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera. Themethod further comprises at the controller, using controller logic,applying the absolute-position-independent relationship between an imagespace placement and a physical space position relative to the camera ina speed estimation algorithm to determine the speed of a vehiclecaptured by the camera.

In accordance with a first exemplary embodiment is provided acamera-based visual speed estimation system for estimating the speed ofvehicles on the basis of video output from of a camera, the camera-basedvisual speed estimation system comprising a controller camera input forreceiving from the video output of the camera comprising at least firstand second images of a travelling vehicle captured at respective firstand second times, the first and second images being defined in an imagespace and the travelling vehicle and camera each belonging to a physicalspace and having respective absolute positions therein. The systemfurther comprises a tangible controller in communication with the camerainput and receiving therefrom the video data from the camera comprisingat least the first and second images, the controller comprisingcalibration logic executable on the controller for causing thecontroller to a) identify in each of the first and second image avehicle feature having a known dimension in the physical space, theidentifying comprising determining a placement of the vehicle feature inthe image space at each of the first and second times; b) define on thebasis of the placement of the vehicle feature in the image space at eachof the first and second times an absolute-position-independentrelationship between an image space placement and a physical spaceposition relative to the camera; and c) apply theabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera in aspeed estimation algorithm to derive speed estimation data indicative ofthe speed of a vehicle captured by the camera. The system furthercomprises a controller output in communication with the controller forreceiving from the controller the speed estimation data and outputtingthe speed estimation data.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be better understood by way of the following detaileddescription of embodiments of the invention with reference to theappended drawings, in which:

FIG. 1 shows a front perspective view of a vehicle speed estimationsystem in accordance with a non-limiting example;

FIG. 2 shows a rear perspective view of the vehicle speed estimationsystem of FIG. 1;

FIG. 3 shows a block diagram of a controller for the vehicle speedestimation system of FIG. 1 according to a non-limiting example;

FIG. 4 shows a perspective view of the camera of the vehicle speedestimation system of FIG. 1 with a physical-space coordinate systemorigin according to a non-limiting example;

FIG. 5 illustrates the consequences of incorrect plate estimation ondisplacement computation on a perspective view of the camera of thevehicle speed estimation system of FIG. 1 according to a non-limitingexample;

FIG. 6 shows an exemplary plate image captured by the camera of FIG. 1with a license plate that is only partially contained within the image ;

FIG. 7 shows exemplary images captured by the camera of the vehiclespeed estimation system of FIG. 1 at two different times overlaid over aperspective view of the camera of the vehicle speed estimation system ofFIG. 1 illustrating the physical position at which these images werecaptured; and

FIG. 8 shows a top plan view of the camera of the vehicle speedestimation system of FIG. 1 with the physical-space coordinate systemorigin of FIG. 4.

FIG. 9 shows a top view of the camera of the vehicle speed estimationsystem of FIG. 1 with a physical-space coordinate system originaccording to a non-limiting example.

DETAILED DESCRIPTION

FIG. 1 shows a vehicle speed estimation system 100 in accordance with anexemplary embodiment. The vehicle speed estimation system 100 comprisesa source of image data providing images of moving vehicles. In this casethe source of image data is a camera 105. The camera 105 is incommunication with a controller 110 that receives image data from thecamera and performs vehicle speed estimation on the basis of the imagedata. FIG. 2 shows the vehicle speed estimation system 100 from thefront.

The camera 105 is a fixed camera used in an LPR system. In particular,in this example the camera is a SharpX™ camera from Genetec™, whichcaptures both color and infrared high-resolution images. In thisexample, speed estimation and calibration will use license platerecognition technology which is better performed using the infraredimages thus only the infrared images will be used for this. The SharpX™camera provides 1024×946 progressive scan frames at 30 frames per secondand has a pulsed LED infrared illuminator allowing it to be used incomplete obscurity. When used in the AutoVu™ system, license platerecognition of reflective plates can occur with plate up to 100 feet (30meters) away.

As shown the camera 105 is in data communication with the controller 110via data communication link 115. The data communication link 115 here isprovided using a cable connection; however other technologies fortransferring image data may be used.

In the illustrated example, the camera 105 and controller 110 aremounted on a pole 101 stemming from a base 102. Advantageously in thisexample the pole 101 is a telescopic pole and the base 102 is a foldablebase allowing the system 101 to be quickly and easily moved from onelocation to another. Although shown here at mid level on the pole, 101,the controller 110 may be mounted high up to prevent tampering. Thecamera 105 is mounted to an adjustable articulation (not shown) allowingthe adjustment of the camera 105's angles with respect to the road so asto allow someone installing the vehicle speed estimation system 100 toframe a portion of the road where vehicles are to be monitored. Also notshown, the vehicle speed estimation system 100 comprises a power sourceproviding power to the controller 110 and, if necessary, the camera 105.This may include a computer power supply-type plug-in device, and/or abattery and/or a solar array.

The installation shown here is merely one of many possibilities. TheSharpX™ and AutoVu™ system is already available for installation onpatrol vehicles. In an alternate embodiment, the controller 110 may bein a vehicle and the camera 105 may be mounted on a sufficientlyelevated roof rack. Advantageously the quick calibration system providedhere allows easy recalibration every time the vehicle, and thus thecamera, is moved to a new location. Other possible installation includesexamples where the camera 105 a security camera mounted, e.g., on abuilding and the controller 105 is implemented on a computer in thebuilding.

The controller 110 of this example comprises a AutoVu™ ALPR processingunit having an Intel™ Atom™ N2600 processor and at least one ALPR camerainput for receiving image data from the camera 105.

Although the present embodiment employs the SharpX™ camera and amodified AutoVu™ ALPR Processing Unit, the scope of the presentcontribution is not intended to be limited to a single platform or setof equipment. Other types of hardware may be used for camera 105, andindeed that source of image data may even be non-camera devices such asbroadcast-receiving equipment receiving image data from elsewhere andproviding it to the controller 110.

FIG. 3 shows a non-limiting example of the vehicle speed estimationsystem 100 and includes a block diagram of the controller 110. As shown,the image data provided by the camera 105 is received at the controller115 at an input 120. The input 120 comprises the logic required torecover images from the image data received from the source of imagedata. In this particular example, camera 105 provides image data in rawformat, however in some embodiments, the camera 105 may comprise an MPEGencoder for encoding the images and transmits the images in encoded formas an MPEG stream. At the controller 110, input 120 may comprise an MPEGdecoder, which is well-known and defined in the various MPEG standards,where the digital stream is decoded to recover images from the imagedata. Naturally other types of decoding and pre-processing logic may beincluded at the input 120.

In alternate embodiments, the image data may be provided to thecontroller 110 in other forms, and the input 120 may be adaptedaccordingly. For example, in one conceivable alternative, the image datamay be provided in analog form and the input 120 may comprise one ormore suitable analog-to-digital converter to recover digital forms ofthe images comprised in the image data.

In this example, an image buffer 125 is provided to buffer incomingimages. Although shown here on its own, the image buffer 125 may beprovided as part of the input 120. The image buffer receives pixelsand/or whole images as they are recovered by the input 120 and storesthem temporarily to release them to the processing logic 130 at the rateat which the processing unit 130 can receive them. Thus the input 120 isin communication with the controller 110 and may be in direct orindirect communication therewith.

The controller 110 comprises a processing logic 130 comprising logic forprocessing the images received in the image data. The processing logic130 of the controller 110 comprises logic modules for performing varioustasks as described herein. In the present embodiment, the processing isperformed by a general-purpose processing unit under instructions fromprogram code stored tangibly in a memory and comprising instructions forthe instructing the general-purpose processing unit to perform thedescribed function. Sufficient detail is provided herein to allow askilled programmer to implement the processing logic without recourse toinventiveness. Other embodiments are possible. In particular, the logicprovided herein may also be implemented as hardware, e.g. in an FPGA orASIC implementation. Here too a skilled hardware programmer is providedsufficient details to implement the taught system in hardware withoutrecourse to inventive impulse.

The controller 110 may include an external memory 140, such as a DDR3memory module serving as random access memory to the processing unit. Aswill be understood, program code of the processing logic may be storedin the memory for loading into the processing unit's program registersto execute the programming logic. However, the programming logic mayalso comprise instructions to store and seek data in memory 140. Inorder to illustrate possible access to the memory 140 by the processinglogic 130, the memory 140 and processing logic 130 are shown as separateentities. This is merely for clarity of the description. The skilledperson will appreciate that the memory 140 may be implemented inmultiple units and/or with multiple layers and details of known memorymanagement and implementation techniques will be omitted herein.

The processing logic 130 is controller logic and comprises logic forperforming various tasks. These are shown here as separate modules. Eachmodule comprises logic, in this case program code instructions toimplement the functionality described herein, which logic may beorganized in any suitable manner, for example in certain programminglanguages a programmer may implement modules in respective classes,however the modules may be implemented in other ways as determined by anon-inventive skilled programmer. The processing logic 130 is thereforeexecutable at the controller for causing the controller to performvarious tasks as described herein.

As shown, the processing logic 130 includes speed estimation logic,shown here as speed estimation logic 136. The speed estimation moduleimplements a vehicle speed estimation algorithm which estimates thespeed of moving vehicles depicted in received images. In the presentembodiment, the speed estimation algorithm requires knowledge of cameraparameters which include the focal length (F), tilt angle (θ_(t)) androll angle (θ_(r)) of the camera 105 which captured the images at thetime at which the images where captured. The present algorithm alsoassumes that the camera position is fixed with respect to the road. Thespeed estimation module 135 computes vehicle speeds on the basis ofdetected movement of a feature of a vehicle that is shown moving in twoor more images at different times. The speed estimation module thereforerequires at least two images of the vehicle in motion, or moreparticularly of the feature used in speed estimation.

In the present embodiment, the feature used for speed estimation is thelicense plate of the vehicle. Advantageously, road vehicles typicallyall have a license plate, the display of which is legally mandatedmaking this feature particularly useful. Moreover license platedimensions are typically known, and in many jurisdictions are constant.For example in most of North America, the width of a license plate oflocal vehicles is known to be 300 mm. Moreover, license plate data mayinclude sufficient information to identify the type of license plate(e.g. a country/jurisdiction code and a license plate number, which mayin turn be used to ascertain the dimensions of the license plate, e.g.by looking it up in a table of license plate dimensions). The licenseplate, as a vehicle feature, has another advantage: it has dimensionsthat are reliably perpendicular to the motion of the vehicle. In thiscontext a dimension is considered to “be” perpendicular to the motion ofthe vehicle if it is measured in a plane or vector that is perpendicularto the motion of the vehicle. For example, the width of the licenseplate is measured along a vector going side-to-side widthwise to thelicense plate, e.g. a vector defining a top edge or a vector defining abottom edge of the license plate and that vector is substantiallyperpendicular to the direction of motion of the vehicle. As such, thewidth is considered herein to be a dimension that is perpendicular tothe motion of the vehicle even though the value of the width, e.g. incentimeters, may itself be a scalar.

Note that while license plates may be installed with a forward ordownward incline, the side-to-side width of the plate will typically beperpendicular to the motion of the vehicle at any given moment. Yetanother advantage of the license plate as a feature is that it isgenerally unmoving in the vertical direction with respect to the road.That is to say, the license plate of a moving vehicle is typically at aconstant height with respect to the road as the vehicle moves. As wewill see later, these benefits allow the implementation of noveltechniques, particularly for calibrating the system and deriving cameraparameters.

For the purposes of describing the processing logic 130, the vehiclefeature used for speed estimation and system calibration will be alicense plate. However, other vehicle features, particularly othervehicle features satisfying the above advantages, may be used invariants of the algorithm. For example where the road-contacting portionof the rear tires can be seen from a satisfactory angle, and if theseare of known distance from one another, the road-contacting portion ofthe rear tire may be used as the vehicle feature with at least thespread between them as known dimension.

The processing logic 130 provides speed estimation data to an output 150which outputs the data in a suitable way from the controller. Theskilled person will appreciate that there are many ways in which speedestimation data may be used, and accordingly, the exact manner andmethod of output may be adapted to a particular use. In the exampleshown in FIG. 1 and FIG. 2, the vehicle speed estimation system 100comprises a wireless communication antenna 155 for communicating speedestimation data wirelessly to a remote station. While the reader willappreciate that this may be done in many ways, in this particularexample the antenna 155 is a wireless phone antenna driven by a modemfor transmitting over a wireless phone data network, such as an LTEnetwork, and the speed estimation data is provided alongside licenseplate recognition data (e.g. the license plate number) and image data(e.g. images of the vehicle showing the license plate) to a remotedevice over an internet connection.

In other embodiments, the output 150 may be an output to a display andmay provide speed estimation data alongside other data visually as amonitor output, e.g. over an HDMI connection. The controller 110 mayimplement a graphical user interface on the monitor in which estimatedvehicle speeds may be displayed numerically, e.g. alongside licenseplate recognition data (e.g. the license plate number) and image data(e.g. images of the vehicle). Output to a monitor does not precludeoutput to a remote device over a network connection, both of which canbe available, e.g. simultaneously.

Thus the speed estimation data received from the processing logic 130 isoutput by the output 150, and may be output in altered form (e.g. in adisplay signal).

The controller 110 may also store speed estimation data, e.g. alongsidelicense plate recognition data (e.g. the license plate number) and imagedata (e.g. images of the vehicle showing the license plate) for latertransmission to an external device. In one alternate embodiment, thecontroller 110 stores this information, e.g. in memory 140, andcomprises a USB connection at which a USB storage device may be dockedand through which the information may be transferred.

When implementing a GUI, the controller 110 may also receive user input,e.g. via GUI manipulations. For example, the controller 110 may provideGUI queries as to whether to start or stop speed estimation (e.g. astart/stop button) and receive user indications in return to start orstop the speed estimation (e.g. a mouse click on that button) and inresponse the controller 110 may start or stop speed estimationoperations. The controller 110 may provide GUI queries as to whether tocalibrate the system (e.g. a calibrate or re-calibrate button) andreceive user indications in return to calibrate the system (e.g. a mouseclick on that button) and in response the controller 110 may calibratethe system. The query as to whether to calibrate the system may take theform of a query as to whether the camera 105 has been moved (e.g. acamera moved button), and the indication to calibrate the system may bean indication that the camera has been moved (e.g. a click on thatbutton).

Advantageously, the vehicle speed estimation system 100 provides anautomatic calibration that does not require any special input from anoperator. Optionally, as described, there may be a “calibrate” commandinput merely to start the calibration, although even this is optional asthe vehicle speed estimation system 100 may be configured, in someembodiments, to run the calibration procedure on every passing vehicleon which the used feature can be recognized, or to run the calibrationprocedure periodically. Alternatively the vehicle speed estimationsystem 100 may be provided with a system for detecting movement of thecamera (e.g. visual detection of movement of the background of theimages from camera 105, or accelerometers on the camera 105) and may beconfigured to run the calibration procedure upon detecting suchmovement.

The speed estimation logic 136 exploits a calibration upon the camera(e.g. done during production of each SharpX™ camera). This calibrationallows transforming a position in an LPR image to a line in space thatis at a known direction with respect to the camera. If the position ofthe camera is known with respect to the road, and the height of thelicense plate is known then it is possible to know the exact physicalposition of the license plate, and of individual identified charactersthereon, with respect to the road.

Knowing the camera position with respect to the road and the plateheight, however, poses certain challenges; however these are determinedby the vehicle speed estimation system 100 in a manner which isexplained herein. With these values found, the speed estimation module135 calculates vehicle speeds. In particular, the speed estimationmodule takes corresponding positions in two images of the same plate(for example, the centers of the same character), maps these imagepositions to physical coordinate system, measures the physicaldisplacement of the positions in the physical coordinate system anddivides this displacement by the time span between these two images.

Here we being to see that there are two difference spaces in which avehicle and a vehicle feature, e.g. the license plate, may be present.The images provided from the camera 105 are defined in an image space inwhich coordinates may be used to define the location of things in theimage. Being two dimensional here, the location of things in an image isdifferent from their location in the physical world. Here the imagespace can be defined by a two dimensional Euclidean coordinate system,specifically an x and y axis running respectively horizontally acrossand vertically across, e.g. from the top left corner of the image,incrementing by a single unit at every pixel. Thus every pixel has aunique coordinate such that objects in an image, such as a license plateor portions thereof, can be defined in the image space by theircoordinates (in this example, their pixel location) in the image space.

On the other hand objects being shown in the image data also in aphysical space in which can be described the real tangible embodiment ofthe object. This physical space can be defined using, e.g. threedimensional Euclidean coordinates. If the physical location of an objectis known at two different times, then the displacement of the object canbe computed. If, in turn the time difference between the two differenttimes is known, or can be computed, then the speed of the object can becomputed by dividing the displacement by the time difference. Now if theobject is travelling linearly, the so-computed speed, in the directionof the displacement, provides an accurate representation of the velocityof the object. FIG. 4 shows an example of physical space coordinatesystem which is described in more details herein.

It is important when determining the time difference between the firstand the second images, when this time difference is used to estimatespeed, to determine this time difference accurately, otherwise theestimated speed may be inaccurate. When the image data is provided by ananalog connection, it may be possible to use synchronization with theVsync signal to obtain time increments between images if the frame rateis known and reliable. However, in the present digital embodiment,counting frames may not be reliable since frames may be dropped ormissed and encoding/decoding processing time may add unknown timevariations. Moreover, using OS-provided time functions may not providesufficient precision since OS thread scheduling may add too muchuncertainty and the system time may be changed by external factors.

In this embodiment, a high accuracy timestamp is generated by the cameradriver or firmware. This timestamp is used to calculate the time spans.The timestamps represent very accurately the time each frame wasacquired and these are provided with the images and once captured arenot affected changes in the OS's time. These timestamps are carried asimage metadata and received at the input 120 and passed on to theprocessing logic 130. These timestamps, being generated in isolation arenot necessarily “absolute” in that they may not necessarily be tied to aspecific date and time. However, subtraction between timestamps is doneand yields time spans between frames as required for speed estimation.

As mentioned above, the height of the license plate is determined inorder to perform speed estimation. However, it is not necessary for anoperator to measure and input an absolute position of any kind for thevehicle, license plate, or any other feature thereof. Rather, theprocessing logic 130 computes the vertical height between the licenseplate and the camera 105. This is sufficient since the license plate cansafely be assumed not to vary in vertical distance from the roadsurface. The geometry is invariant to a vertical translation that isdone together for the camera and the license plate.

Incorrect height estimation can lead to incorrect speed estimation. Inone example illustrated in FIG. 5, estimating incorrectly estimating alower height for the license plate lead to the computation of adisplacement that is longer than the actual displacement of the vehicle,leading to an overestimation of the speed of the vehicle.

The system employs knowledge of a known dimension to estimate thefeature, in this case the license plate, height. One very reliabledimension for North American vehicles is the plate width. In virtuallyall jurisdictions of North America, the plate is fairly precisely 300 mmwide. Even in the presence of a frame around the plate, the width isonly reduced by a few millimeters.

In order to compute feature height, the location of the feature inimages is identified. To this end, the processing logic comprises imagefeature identification logic, here in the form of image featureidentification module 160. The Image feature identification modulereceives two or more images from the image data each showing at leastpartially a same vehicle, and showing a particular feature, in this casethe license plate, of the vehicle. The image feature identificationmodule 160 comprises controller logic executable on the controller 110to cause the controller to determine a placement of the vehicle featurein the image space at each of the first and second times. A placementmay include various measures defining the location and/or dimensions ofthe vehicle feature, and in the present example the placement includesthe image-space coordinates of at least a portion of the vehiclefeature.

For each of the at least two images, the image feature identificationmodule 160 performs a license plate recognition algorithm to extract theimage-space location (specifically here, pixel coordinates) of thelicense plate, and in particular of the left side and right side of thelicense plate in each image. The image feature identification module 160performs an extraction of a license plate region, a segmentation andrecognition of the license plate features, in this case license platecharacters (e.g. for logging purposes) and the left and right edges ofthe license plate to derive the image-space location (specifically here,pixel coordinates) of the license plate, and in particular of the leftside and right side of the license plate in each image. For the left andright edges, various specific points may be selected to derive adiscrete coordinate address, such as top or bottom corners. In thepresent example, points on the left and right edges are selected at theheight of the bottom of each character (i.e. through an imaginary lineat the bottom of the license plate characters) as ascertained by the LPRdetection.

The feature identification module 160 accesses the image space locationof the license plate in the images to obtain the portion of the imagesfeaturing the license plate. The plate height module 161 comprises ade-rotation and de-skewing algorithmic logic which it applies to thelicense plate image in order to generate a plate image that is a flat(horizontal) rectangle. The feature identification module 160 comprisesalgorithmic logic (in this case program code instructions) to calculatethe sum of the image intensity (Y value) columns of the image andapplies it to generate an image intensity profile. The featureidentification module 160 further comprises algorithmic logic to computethe absolute derivative of the image intensity profile, and to identifythe positions of the maximum absolute derivative and applies it toderive, based on the left side and right side positions of maximumabsolute derivative the left side and the right side of the plate, whichare where the intensity profile varies the most.

Processing logic 130 comprises plate height computing logic, here in theform of plate height module 161.

If the following are known:

position of the camera with respect to the road

position of the plate corners in the source image

physical width of the license plate (in this case, the width of 300 mmis used) Then, it is possible to calculate the height of the licenseplate by solving the equation that imposes that the lines originatingfrom the camera, in the direction of a point on the left side of theplate and in the direction on the right side of the plate, are distantof the physical width of the plate when they intersect the plane at theplate height.

The processing logic 130 assumes a set of physical constraints. In thisexample, the set of physical constraints comprises a known dimension ofthe vehicle feature, in this particular case the width, which is knownto be 300 mm. In this example the set of physical constraints furthercomprises that the known dimension is perpendicular (that is to say thatit is measured in a direction that is perpendicular) to the direction oftravel of the vehicle. In this example the set of physical constraintsfurther comprises that the vehicle feature maintains a constant verticaldistance from the camera. The processing logic 130 assumes theseconstraints either by providing such constraints as/on variables in acomputation or by applying computations that require these constraintsto hold, as is done in the example provided herein.

In this calculation, the license plate is furthermore always consideredto be below the camera 105, with respect to the road.

In order to prevent errors, the plate height module 161 imposes performsa discrimination on images and uses only images that conform to acertain standard of quality. In this example, the plate height module161 does not estimating the plate height from the road using imageswhere the plate is very close to the borders of the image since in theseimages, the plate can be cropped, as is the case in the image shown inFIG. 6, leading to wrong points in the source image. This discriminationmay also take place at the feature identification module 160, which maydiscount images where the plates are too close (e.g. closer than apreset threshold) to the borders of the image.

This discrimination may be applied to the speed estimation and/or thecalibration process as a whole in order to avoid incorrect speedestimation or camera parameter estimation. In this example, speedestimation is not performed if all images for a plate are likely to becropped (e.g. they are all close to the image border), if not all imagesfor the same plate give approximately the same height, or if thecalculated plate height is very unlikely (e.g. below the road, or over1.5 m from the road).

The speed estimation module 135 comprises logic for deriving thephysical space position of a feature of an image on the basis of itsimage space position, logic for computing a travel distance over aplurality of images and logic for computing speed of travel. Inparticular, the speed estimation module uses anabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera in aspeed estimation algorithm to determine the speed of vehicles capturedby the camera. The absolute-position-independence means that knowledgeof the precise location of the camera with respect to the physicalenvironment, e.g. the road, is not required as image-space placements(e.g. position) are related only to physical-space positions relative tothe camera.

In this particular example, the speed estimation module 135 convertsimage space coordinates to physical space coordinates using the tiltangle, roll angle and focal length. The speed estimation module 135comprises logic, in this example in the form of computer-readableinstructions for computing the matrix that transforms a coordinate froma coordinate system aligned with the camera itself to the physical spacecoordinate system as follows:

$\begin{matrix}{M = {\begin{pmatrix}{- 1} & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 1\end{pmatrix}\begin{pmatrix}1 & 0 & 0 \\0 & {\cos \; \theta_{t}} & {\sin \; \theta_{t}} \\0 & {{- \sin}\; \theta_{t}} & {\cos \; \theta_{t}}\end{pmatrix}\begin{pmatrix}{\cos \; \theta_{r}} & {\sin \; \theta_{r}} & 0 \\{{- \sin}\; \theta_{r}} & {\cos \; \theta_{r}} & 0 \\0 & 0 & 1\end{pmatrix}}} \\{= \begin{pmatrix}{{- \cos}\; \theta_{r}} & {{- \sin}\; \theta_{r}} & 0 \\{\sin \; \theta_{t}\sin \; \theta_{r}} & {{- \sin}\; \theta_{t}\cos \; \theta_{r}} & {\cos \; \theta_{t}} \\{{- \cos}\; \theta_{t}\sin \; \theta_{r}} & {\cos \; \theta_{t}\cos \; \theta_{r}} & {\sin \; \theta_{t}}\end{pmatrix}}\end{matrix}$

The image space to physical space conversion module 162 furthercomprises logic to compute the function ƒ(X,z)=(ƒ_(x)(X,Y, z),ƒ_(x)(X,Y, z), z) that converts the position of a point X=(X,Y) in imagecoordinate system to a point x=(x,y,z) in the physical space coordinatesystem:

${f_{x}\left( {X,Y,z} \right)} = {z\frac{{{M_{11}\left( {X - c_{X}} \right)}p_{x}} + {{M_{12}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{13}F}}{{{M_{31}\left( {X - c_{X}} \right)}p_{x}} + {{M_{32}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{33}F}}}$${f_{y}\left( {X,Y,z} \right)} = {z\frac{{{M_{21}\left( {X - c_{X}} \right)}p_{x}} + {{M_{22}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{23}F}}{{{M_{31}\left( {X - c_{X}} \right)}p_{x}} + {{M_{32}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{33}F}}}$

where

-   c_(x) and c_(y) are the pixel coordinates of the image center-   p_(x) and p_(y) are the physical dimensions of the camera pixels-   F is the focal length-   M_(ij) is the i-th line, j-th column element of matrix M.

As will be appreciated further on, some of this logic described hereperforms functions similar to logic in the calibration logic and forexample the physical space conversion module 162. Accordingly, ratherthan to provide new logic in the speed estimation module 135, the speedestimation module may simply make use of logic provided in thecalibration logic 170, which logic may be included in the speedestimation logic as well.

The speed estimation module 135 comprises logic to derive from thephysical location of the license plate in two or more images thedisplacement undergone between the two or more images. To that end, thespeed estimation module 135 may comprise logic to compute a differencevector between each position. Based on the difference vector, the speedestimation module 135 comprises logic for determining a distancetravelled. The speed estimation module 135 also comprises logic forderiving a time difference between each position, in this example byusing time stamp information (e.g. subtracting the time conveyed by thetimestamp associated with the later image from the time conveyed by thetimestamp associated with the earlier image). By dividing the distancetravelled with the time difference during travel, for which the speedestimation module 135 comprises logic, the speed estimation module 135can derive a speed estimate for the vehicle. If multiple images areused, multiple distance vectors can be computed to derive multipletravel distances each corresponding to multiple time differences. Thesemay be used to derive a more accurate speed estimation, for example bytaking the average of individually computed speeds.

Thus the vehicle speed estimation system 100 derives estimated speedsfor vehicles on the basis of the image data from the camera 105. Asmentioned, camera position is used in the estimation. Advantageously,the vehicle speed estimation system 100 comprises logic for determiningcamera parameters on the basis of the image data. This calibration ofthe system can be done in some embodiments with little or no humanintervention. In one embodiment, the calibration yields a relationshipbetween the image space and the physical space that allows thederivation of a position of a feature in the physical space on the basisof the position of the feature in images in the image space.

The processing logic 130 comprises calibration logic 170 which performssystem calibration. Calibration logic 170 comprises several modules oflogic described herein as well as logic for applying or using thesemodules. In the present example, calibration logic 170 also comprises acalibration module 171 which controls the calibration process andcomprises, amongst other logic calling for use of the other modules usedin calibration. In one software example the calibration module 171comprises instructions calling instances of the modules other modulesused in calibration (if these modules are implemented, for example, asinstantiable classes) to perform their particular task with parametersprovided by the calibration module 171. The modules that are used by thecalibration logic 170 are shown here as being inside the calibrationlogic 170. This is to indicate that they together form the logic used incalibration, however this does not necessarily mean that they must behierarchically encapsulated by the calibration logic 170. For example,some or all modules may be implemented e.g. by software classes that areindependent from, but callable by, the calibration module 171's ownlogic (which may itself in one example be defined by a class of itsown). When these modules are used in this manner, they together form thelogic that performs calibration. Note, however that some of the modulesmay be used for other purposes, for example the plate height module 161is also used in speed estimation and thus also forms part of the speedestimation logic 136 which comprises the speed estimation module 135 andthe plate height module 161.

Calibration logic 170 includes logic for determining camera parameters.The calibration logic 170 takes advantage of the fact that the plate hasa displacement that is parallel to the road (i.e. the plate's heightwith respect to the road is generally constant), and that license platescomprise a dimension (e.g. the width) that is parallel to the road andthat is perpendicular to the direction of motion. As withperpendicularity to the direction of travel, the dimension is consideredherein to be parallel to the road if it is measurable in a vector orplane that is parallel to the road. In this particular case the left andright sides of the plate are at the same height with respect to the roadand the width is measured along a line that itself is parallel to theroad. The plates may be tilted frontwardly or rearwardly but their widthremains perpendicular to the motion of their respective vehicles. Thecalibration logic 170 also takes advantage of the fact that licenseplates have a displacement that is perpendicular to their bottom border,that the positions of the corners of the plates can be found asdescribed herein, and that the physical width of the plates can beknown. The calibration logic 170, takes advantage of these things andapplies an assumption that these facts are true. In this example thecalibration logic 170 also makes the additional assumptions that somecamera parameters (in this case the focal length, F) is/are known andthat the camera is positioned at a height above that of the licenseplates in the images it captures and that it looks downwards towards theroad.

However, the calibration logic 170 performs calibration without priorknowledge of the camera installation height, the camera pan angle(leftward, rightward), the camera downward tilt angle (θ_(t)), thecamera roll angle about its optical axis (θ_(r)), the physical height ofeach license plate with respect to the road, the trajectory and speed ofthe vehicle with respect to the road, and the focal length, if it hadnot been measured prior to installation.

Advantageously, the vehicle speed estimation system 100 applies ageometry that is invariant to a vertical translation that is donetogether for the camera and the license plates, and as such, the actualphysical height of the camera is not required. The physical space isdefined in terms of coordinates that are centered not on the groundelsewhere but in relation to the ground or ground-based location, butrather on the camera 105, or elsewhere but in relation to the camera105. This also resolves the problem of angled roads. Since the geometryof the physical space is not defined based on the ground itself, it isnot necessary for the ground to be horizontal, i.e. not going uphill ordownhill. For the same reason, thanks to the calibration model used, theactual height of each license plate with respect to the road isirrelevant and does not need to be known or estimated. Thus it is notnecessary to use a special vehicle with a known plate height tocalibrate the system. With the calibration model used, the trajectoryand speed of the vehicle with respect to the road are also irrelevant,thus allowing the system to use any passing vehicle, rather than acontrolled vehicle to calibrate the system

Moreover, the vehicle speed estimation system 100 is robust tomiscalculations or false assumptions with respect to the pan angle sincethe geometry is invariant to the camera pan angle. If the pan angle iswrongly estimated, the plates calculated trajectories may not beparallel to the road, but the estimated speed will still be correct.

The calibration logic 170 performs the calibration described herein andthen makes an assumption as to the camera installation height. In thisparticular example, the calibration logic 170 takes the height of one,or of an average of more than one, license plates and simply assumes areasonable height for the license plate, in this example 600 mm. Thecalibration logic 170 then adds the computed height of the plate withrespect to the camera (an absolute value equivalent to the height of thecamera with respect to the plate) to this plate height and assumes thisto be the physical height of the camera with respect to the road. Thecalibration logic 170 may then store this value in a memory location foruse, e.g., by the speed estimation module 135. Although the height oflicense plates and/or of the camera with respect to the road are notrequired for speed estimation, the camera height is optionally used toimplement error checking. In one example vehicle speed estimation is notperformed if the license plate height is detected as being below theroad level.

The calibration logic 170 similarly makes an assumption regarding thepan angle. Specifically, the calibration logic 170 computes thedisplacement for one license plate, or the average displacement for morethan one license plate, and sets a pan angle such that the displacementdirection is parallel average displacement (average displacement isgenerally parallel to the road). The calibration logic 170 may thenstore this value in a memory location for use, e.g., by the speedestimation module 135.

Although the calibration logic 170 derives assumed values for cameraheight and pan angle, accurate speed estimation can be performed withonly the camera downward tilt angle θ_(t), the camera roll angle aboutit optical axis, θ_(r), and the focal length.

In this example, the focal length is a known constant that is acharacteristic of the camera used. As described herein, the calibrationlogic 170 determines the tilt and roll angles. The logic implied,described in full details herein, may involve assuming a hypotheticaltilt and roll angle, calculating the vertical distance of at least onelicense plate in at least two images with respect to the camera in themanner taught herein, deriving the position in the physical space of thelicense plate in each image, converting the position in the physicalspace back to positions in the image space, computing an error betweenthe so-computed position in the image space and the actual position ofthe license plate in the image space of the images used, modifying thehypothetical tilt and roll angle on the basis of the error and repeatingthe process until no more reduction of the error is possible. Theassumed hypothetical tilt and roll angles are then considered to be thecamera 105's actual tilt and roll angles and the calibration logic 170may then store these value in a memory location for use, e.g., by thespeed estimation module 135.

To this end, the calibration logic 170 employs the image featureidentification module 160 as described herein to identify a particularlicense plate in at least two images (corresponding respectively to afirst and second time) and determine the license plate placement at eachof the first and second times. In this particular example, thecalibration logic 170 first identifies in an image stream all the imagesin a sequence that have a same license plate and that satisfy thediscrimination described herein. It then selects the first and last suchimage where the license plate appear. In these selected two images, theimage feature identification module 160 identifies the location of apoint on the left side of the license plate in the first image in themanner described herein. This point will be referred to as Point Aherein. The image feature identification module 160 identifies thelocation of a point on the right side of the license plate in the firstimage in the manner described herein. This point will be referred to asPoint B herein. The image feature identification module 160 identifiesthe location of a point on the left side of the license plate in thesecond image in the manner described herein. This point will be referredto as Point C herein. And finally, the image feature identificationmodule 160 identifies the location of a point on the right side of thelicense plate in the second image in the manner described herein. Thispoint will be referred to as Point D herein. FIG. 7 illustrates pointsA, B, C and D in one particular example.

These positions of these points with respect to the road are unknown.However, because of some of the assumptions made by the calibrationmodule 170, these positions are not completely independent from eachother. In particular, the assumptions mean that the four points have thesame height, that the distance between A and B and the distance betweenC and D are equal to the physical width of the license plate, and thatsince the plate is assumed to travel perpendicularly to its long end,points A, B, D and C, when translated to the physical space, form arectangle.

It is possible to express the positions of the points A, B, C, D usingonly the following 5 unknown quantities: (x_(A), y_(A), z_(A), x_(c),y_(A)). If they become known, these quantities (along with thehypothesis listed above) would completely determine the positions of allthe points. The other unknown quantities are the tilt angle and the rollangle of the camera. The focal length may also unknown if it had notbeen measured prior to installation.

Each license plate provides 8 equations to help determining theseunknown quantities. Given physical space coordinates for points A, B, Cand D, these coordinates can be converted using the roll angle, tiltangle and focal length camera parameters to the image space coordinates.If the camera parameters used are accurate, then the four points A, B,C, D so converted should be found to be in the same position as wasidentified by the image feature identification module 160.

In short, for a passing license plate, we have 5 n+2 (+1) unknownquantities (x_(A), y_(A), z_(A), x_(c), y_(c) of each plate, tilt androll, possibly focal length) and 8 n equations (4 points, x and y). Thusonly a single plate is necessary to have enough equations to determineall the quantities. However, more may be used in order reduce the impactof an error in the computations related to a particular plate. Thealgorithm described herein minimizes a sum of errors. More plates/imagesmay be used by minimizing the errors over all the image positions of allthe plates. To this end, the calibration logic 170 may operate on anumber of plates, e.g. 20. It waits until all the plates have passed,then taking all the position data of the plates in all the images it hasselected it performs the calibration by minimizing the error as taughtherein iteratively until no more error reduction can be found. For thepurpose of clarity of description, the process is described for a singleplate taken at two positions (in two images).

Camera parameters are estimated by the calibration logic 170 asdescribed herein. Herein, bold indicates a vector notation where a lowercase x=(x, y, z) denotes physical space coordinates and upper case X=(X,Y) denotes image space coordinates.

If multiple plates are used in the calibration process, the featureidentification module 160 identifies the image space positions of leftand right points of the plates in the first and in the last images whereeach plate i (i=1 . . . n) appears. These points are expressed in pixelcoordinates in the images, where the axis origin is centered on the topleft pixel, with the X axis going rightward and the Y axis goingdownward in the image. Thus, we derive for each plate:

-   -   X_(iA)=(X_(iA), Y_(iA)) the coordinates of the left point in the        first image of the plate i    -   X_(iB)=(X_(iB), Y_(iB)) the coordinates of the right point in        the first image of the plate i    -   X_(iC)=(X_(iC), Y_(iC)) the coordinates of the left point in the        last image of the plate i    -   X_(iD)=(X_(iD), Y_(iD)) the coordinates of the right point in        the last image of the plate i

The calibration logic then identifies the plate width w_(i) (expressedhere in millimeters) of each of the plate i. In this particular example,the vehicle speed estimation system 100 is configured for use in NorthAmerica in jurisdictions where w_(i) is constant and its value is 300mm. Therefore this value is a pre-set constant, e.g. stored as aconstant variable. However, in alternate embodiments, the calibrationlogic 170 may comprise logic for identifying the plate width on thebasis of detected plate features such as a detected country code and/orplate number, e.g. by looking these plate features up in a database ofplate widths.

For the purposes of the present description of this example, thephysical space coordinates are defined as shown in FIG. 4, as having anorigin centered on the camera 105's focal point, with the x and y axisparallel to the road plane, the z axis pointing downward and the x and yaxis oriented such than the camera has a pan angle of 0. This coordinatesystem is preferred for simplicity of the computations required herein,however, other coordinate systems defined about, or in relation to, thecamera 105 may be used, although this requires adaptation of thecomputations to the other coordinate system. FIG. 4 shows a perspectiveview of the physical state coordinate system axes, while FIG. 8 showsthe same in a top view.

As mentioned herein, in this example the focal length F is a knownconstant that is a camera characteristic. However, if the focal length Fis not known, the calibration sets it to an arbitrarily large value. Thecalibration logic 170 may store this value in a memory location, in thiscase as a variable.

The calibration logic also sets arbitrary, but plausible, values forcamera tilt angle θ_(t) (downward toward the road) and roll angle θ_(r)(around its optical axis). The calibration logic 170 may store thisvalue in a memory location, in this case as a variable. These initialvalues may in fact be pre-set.

The calibration logic 170 comprises logic for deriving the physicalspace position of a feature of an image on the basis of its image spaceposition. In particular, image space position to physical space positionconversion logic is provided in the image space to physical spaceconversion module 162 which converts image space coordinates to physicalspace coordinates using the tilt angle, roll angle and focal length. Theimage space to physical space conversion module 162 comprises logic, inthis example in the form of computer-readable instructions for computingthe matrix that transforms a coordinate from a coordinate system alignedwith the camera itself to the physical space coordinate system asfollows:

$\begin{matrix}{M = {\begin{pmatrix}{- 1} & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 1\end{pmatrix}\begin{pmatrix}1 & 0 & 0 \\0 & {\cos \; \theta_{t}} & {\sin \; \theta_{t}} \\0 & {{- \sin}\; \theta_{t}} & {\cos \; \theta_{t}}\end{pmatrix}\begin{pmatrix}{\cos \; \theta_{r}} & {\sin \; \theta_{r}} & 0 \\{{- \sin}\; \theta_{r}} & {\cos \; \theta_{r}} & 0 \\0 & 0 & 1\end{pmatrix}}} \\{= \begin{pmatrix}{{- \cos}\; \theta_{r}} & {{- \sin}\; \theta_{r}} & 0 \\{\sin \; \theta_{t}\sin \; \theta_{r}} & {{- \sin}\; \theta_{t}\cos \; \theta_{r}} & {\cos \; \theta_{t}} \\{{- \cos}\; \theta_{t}\sin \; \theta_{r}} & {\cos \; \theta_{t}\cos \; \theta_{r}} & {\sin \; \theta_{t}}\end{pmatrix}}\end{matrix}$

The image space to physical space conversion module 162 furthercomprises logic to compute the function ƒ(X,z)=(ƒ_(x)(X,Y, z),ƒ_(x)(X,Y, z), z) that converts the position of a point X=(X, Y) inimage coordinate system to a point x=(x, y, z) in the physical spacecoordinate system:

${f_{x}\left( {X,Y,z} \right)} = {z\frac{{{M_{11}\left( {X - c_{X}} \right)}p_{x}} + {{M_{12}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{13}F}}{{{M_{31}\left( {X - c_{X}} \right)}p_{x}} + {{M_{32}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{33}F}}}$${f_{y}\left( {X,Y,z} \right)} = {z\frac{{{M_{21}\left( {X - c_{X}} \right)}p_{x}} + {{M_{22}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{23}F}}{{{M_{31}\left( {X - c_{X}} \right)}p_{x}} + {{M_{32}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{33}F}}}$

where

-   c_(x) and c_(y) are the pixel coordinates of the image center-   p_(x) and p_(y) are the physical dimensions of the camera pixels-   F is the focal length-   M_(ij) is the i-th line, j-th column element of matrix M.

The calibration logic 170 also comprises logic for deriving the imagespace position of a feature of an image on the basis of its physicalspace position. In particular, physical space position to image spaceposition conversion logic is provided in the physical space to imagespace conversion module 164 which converts image space coordinates tophysical space coordinates using the tilt angle, roll angle and focallength. To this end the physical space to image space conversion module164 comprises logic that computes the function ƒ⁻¹(x)=(ƒ_(x) ⁻¹(x,y,z),ƒ_(y) ⁻¹(x,y,z)) that converts the position of a point x=(x, y, z) inthe physical space coordinate system to a point X=(X, Y) in imagecoordinate system:

${f_{X}^{- 1}\left( {x,y,z} \right)} = {{\frac{F}{p_{x}}\frac{{M_{11}^{- 1}x} + {M_{12}^{- 1}y} + {M_{13}^{- 1}z}}{{M_{31}^{- 1}x} + {M_{32}^{- 1}y} + {M_{33}^{- 1}z}}} + c_{X}}$${f_{Y}^{- 1}\left( {x,y,z} \right)} = {{\frac{F}{p_{y}}\frac{{M_{21}^{- 1}x} + {M_{22}^{- 1}y} + {M_{23}^{- 1}z}}{{M_{31}^{- 1}x} + {M_{32}^{- 1}y} + {M_{33}^{- 1}z}}} + c_{Y}}$

where

-   M_(ij) ⁻¹ is the i-th line, j-th column element of matrix M⁻¹ that    is the inverse of matrix M.

The calibration logic 170 comprises logic executable on the controller110 to cause the controller 110 to assume an assumed set of cameraparameters. The set of camera parameters may include a value of Focallength F, camera tilt angle θ_(t) and roll angle θ_(r), however in thisexample the assumed set of camera parameters include only camera tiltangle θ_(t) and roll angle θ_(r).

In a calibration process, after having derived the image space locationof the four points A, B, C, and D of at least one license plate at theimage feature identification module 160, the rough location of each ofthe license plate in the physical space coordinate system is computed bythe image space to physical space conversion module 162 using thecurrent values of the tilt angle, roll angle and focal length, which ina first iteration are the preset values or values set by the calibrationlogic 170 described above (arbitrary, but plausible, values for cameratilt angle θ_(t) and roll angle θ_(r) and the focal length of the cameraor an arbitrary large value) and enforcing the license plate width fromthe first image. This assumed set of camera parameters is a first set ofcamera parameters from a plurality of sets of camera parameters. Thecalibration logic 170 then causes the controller 110 to define atentative absolute-position-independent relationship between image-spaceplacement and physical-space position on the basis of the assumed set ofcamera parameters.

To this end, the calibration module 170 begins by setting the licenseplate z coordinate to be at a test value z_(test) (for examplez_(test)=1000 mm), which it may store in memory although which in afirst instance may be a preset value. Using this value, the calibrationmodule 170 causes the image space to physical space conversion module162 to compute the positions of the license points A (left point infirst image) and B (right point in first image) in the physical space ifthese points were at z_(test):

x _(test iA)=ƒ(X _(iA) , z _(test))

x _(test iB)=ƒ(X _(iB) , z _(test))

The calibration module 170 also comprises positional informationcomputation logic provided here in positional information computationmodule 163. The positional information computation module 163 compriseslogic to perform geometric computation to compute positional informationand dimensions in three dimensions on the basis of physical spacepositional information and more specifically in this case on the basisof physical space coordinates.

After having computed the position of license points A and B asdescribed immediately above, the calibration module 170 causes thepositional information computation module 163 to compute the width ofeach license plate if the plates were at z_(test):

w _(test i) =|x _(test iB) −x _(test iA|)

That being done, the positional information computation module 163 thencomputes the z coordinate of each license plate such that the licenseplates have the proper width:

z _(i) =w _(i) /w _(test i)

That being done, the calibration module 170 then causes the image spaceto physical space conversion module 162 to compute the physical spacepositions of the license points A (left point in first image) and C(left point in last image) according to this z coordinate:

x _(iA)=ƒ(X _(iA) , z _(i))

X _(iC)=ƒ(X _(iC) , z _(i))

Thus the tentative absolute-position-independent relationship is used tocompute a physical space location (which in this case is a relativephysical space location, relative to the camera) of at least a portionof the vehicle (in particular at least a portion of the vehicle feature,in this case points A and C). This is done at each of the first andsecond times (e.g. point A at the first time and point C at the secondtime). This will be used in evaluating the validity of the tentativeabsolute-position-independent relationship, and in particular inverifying compliance with the set of physical constraints, that is tosay with at least one of the physical constraints assumed.

The calibration module 170 then causes positional informationcomputation module 163 to complete the computation of positions of the 4points of the license plates by enforcing that the plate has the properwidth and that its motion is perpendicular to its top and bottom sides.To this end, the positional information computation module 163 compriseslogic for computing a displacement vector which it applies to computethe displacement vector:

d _(i) =x _(iC) −x _(iA)

The positional information computation module 163 comprises logic forcomputing the vector parallel to the plate top and bottom sides with thelength equal to the width of the license plate which it applies:

e _(i) =w _(i)(0, 0, 1)×(d _(i) /|d _(i|))

The positional information computation module 163 may if necessary,negate this vector such that its x component is always negative.

Thus we now have the vector that defines the spatial relationshipbetween the opposite sides of the license plate if points A, B, C and Dform a rectangle as they should. The positional information computationmodule 163 also comprises logic to compute the physical space positionsof the points B (right point in first image) and D (right point in lastimage) based on this vector and applies it:

x _(iB) =x _(iA) +e _(i)

x _(iD) =x _(iC) +e _(i)

Now the calibration module 170, once this is done, causes the physicalspace to image space conversion module 164 to compute the predictedpositions of these 4 points in the image coordinate system:

X′ _(iA)=ƒ⁻¹(x _(iA))

X′ _(iB)=ƒ⁻¹(x _(iB))

X′ _(iC)=ƒ⁻¹(X _(iC))

X′ _(iD)=ƒ⁻¹(x _(iD))

The calibration logic 170 also comprises an error computation module 165which comprises logic for computing an error between different positionsin a given space. The calibration logic 170 comprises logic executableat the controller 110 for causing the controller 110 to evaluate thevalidity of the tentative absolute-position independent relationship. Ifthe validity is established, this tentative absolute-positionrelationship may be established as the selected one with which toperform speed estimation. In this example, since the assumptions areapplied and image space positions are computed based on the assumptions,the error computation module 165 comprises logic for computing errorsbetween different positions in the image space, and more particularlyhere for computing the pixel errors between the measured pixel positionsand the predicted pixel positions for all the points of all the licenseplates. After the predicted positions of the four points have beencomputed as described, the calibration logic causes the errorcomputation module 165 to compute the pixel errors between the measuredpixel positions and the predicted pixel positions for all the points ofall the license plates:

E _(iA)=(E _(X iA) , E _(Y iA))=X′ _(iA) X _(iA)

E _(iB)=(E _(X iB) , E _(Y iB))=X′ _(iB) X _(iB)

E _(iC)=(E _(X iC) , E _(Y iC))=X′ _(iC) X _(iC)

E _(iD)=(E _(X iD) , E _(Y iD))=X′ _(iD) X _(iD)

The error computation module 165's logic also computes the total squaredpixel error:

E ²=Σ_(i=1) ^(n)(|E _(iA)|² +|E _(iB)|² +|E _(iC)|² +|E _(iD)|²)

We thus have an error value that is caused by inaccuracies in theassumptions made, and more specifically by inaccuracies in the selectedvalues of camera parameters, namely the tilt angle, roll angle and focallength (if it wasn't known).

This having been established, the calibration module 170 may store in amemory an indication of the error value obtained for the particularvalues of camera parameters used. The calibration module 170's logicimplements an iterative approach whereby different parameters sought bythe calibration process, in this case the camera parameters, and morespecifically in this case the tilt angle θ_(t) and the roll angle θ_(r)are tested to see their impact on the computed error the assumption ofthese parameters yields. If the evaluation of the tentativeabsolute-position-independent relationship indicates that the tentativerelationship is valid, it may be established as the selectedrelationship with which to perform speed estimation. In practice, it maybe required to try several sets of camera parameters before validity canbe established. Thus the steps of assuming an assumed set of cameraparameters, defining a tentative absolute-position-independentrelationship between and image space placement and a physical spaceposition relative to the camera, evaluating the validity of using thetentative relationship to compute a physical space location of thevehicle feature and verifying compliance with the assumed physicalconstraints may be repeated several times. In particular, it may berepeated until a set of camera parameters that provides the lowest erroris found, the resulting relationship being selected.

More specifically, the calibration module 170 comprises logic to now seta new value for the tilt angle θ_(t) and the roll angle θ_(r), as wellas the focal length F if it isn't known, and cause the repetition of thesteps described above subsequent to the setting of θ_(t) and θ_(r) (andoptionally F) to derive a new error value corresponding to these newcamera parameters. This is then repeated, under the calibration logic170's logic, for all plausible values, using small increments (forexample: θ_(t) varying from 0° to 70° with 1° steps and θ_(r) varyingfrom −20° to 20° with 1° steps, and (if F is uknown) varying F from 8 mmto 60 mm in 1 mm steps), each time storing the error associated with thecamera parameters used.

Comparison logic, shown here as its comparator module 166 but which maybe implemented directly in the calibration module 170's logic, comparesthe different errors found to identify the values of θ_(t) and θ_(r)(and F, if unknown) that yield the smallest total squared pixel error Eand take note of the physical space coordinates x_(iA) and x_(iC) of thepoints A and C for all license plates when computed at these values ofθ_(t), θ_(r) and F. (To this end the values of x_(iA) and x_(iC) may bestored, e.g. by the calibration logic 170, alongside the errors E or thevalues of x_(iA) and x_(iC) can be recomputed. While the comparisonlogic may be implemented after all errors have been computed, otherimplementations are possible, such as a case-by-case comparison (uponcomputation of each error) of each error with the smallest error, withthe computed error replacing the smallest error if it is smaller.

We now have estimated values of the camera parameters θ_(t), θ_(r) andF. This estimate can be further refined using an optimization processthat is more uniform in its treatment of the 4 points of each licenseplate.

To this end, the calibration module 170 comprises refinement logic, inthis example provided in refinement module 167 that applies anoptimization to the estimation of the camera parameters to derive moreaccurate camera parameters. The refinement module 167's logic alsocomputes the focal length if it was not known.

The calibration module 170 forms a parameter vector with the followingvariables which is provided to the refinement module 167 to optimize.The vector is initialized with the following parameters:

θ_(t) (which is the tilt angle found by the process described above)

θ_(r) (which is the roll angle found by the process described above)

F

x_(iA)

y_(iA)

x_(iC)

y_(iC)

z_(i)

The last 5 variables to optimize exist for each license plate, so thetotal parameter vector length is 5n+3.

A function is defined that computes X and Y pixel errors for each of the4 points of each of the license plate according to the values of theparameter vector to optimize. In this example, this function employs thephysical space to image space conversion module 164 to convert thephysical space parameters to image space parameters as described aboveand employs the positional information computation module 163 to computethe physical space position of the points B (right point in first image)and D (right point in last image) based on the values of x_(iA) andx_(iC) provided in the parameter vector:

x _(iB) =x _(iA) e _(i)

x _(iD) =x _(iC) e _(i)

Now the refinement module 167's logic, once this is done, causes thephysical space to image space conversion module 164 to compute thepredicted positions of these 4 points in the image coordinate system:

X′ _(iA)=ƒ⁻¹(x _(iA))

X′ _(iB)=ƒ⁻¹(x _(iB))

X′ _(iC)=ƒ⁻¹(x _(iC))

X′ _(iD)=ƒ1 ⁻¹(x _(iD))

The refinement module 167's logic then causes the error computationmodule 165 to compute the pixel errors between the measured pixelpositions and the predicted pixel positions for all the points of allthe license plates:

E _(iA)=(E _(X iA) , E _(Y iA))=X′ _(iA) −X _(iA)

E _(iB)=(E _(X iB) , E _(Y iB))=X′ _(iB) −X _(iB)

E _(iC)=(E _(X iC), E_(Y iC))=X′ _(iC) −X _(iC)

E _(iD)=(E _(X iD), E_(Y iD))=X′ _(iD) −X _(iD)

The refinement module 167 also comprises logic for applying theLevenberg-Marquardt algorithm, and applies it to jointly find theoptimal values of the parameter vector such that the total squared erroris minimized.

At the end of the optimization process, the resulting values of θ_(t)and θ_(r) form the final estimate of the tilt and roll angles of thecamera 105. If desired logic may be provided to do a transformation ofthe physical coordinate system, for example such that the origin is at adifferent position, e.g. underneath the camera at the road level withits y axis direction parallel to the direction of the road. For this,optional logic may be provided to the calibration module 170 to computethe camera installation height as

$h = {{\frac{1}{n}{\sum\limits_{i = 1}^{n}z_{i}}} - {600\mspace{14mu} {mm}}}$

where 600 mm is an arbitrary plausible average height of the licenseplates with respect to the road and to compute the total displacementvector of all the plates:

$d = {\sum\limits_{i = 1}^{n}d_{i}}$

(in this summation, some of the d_(i) may be negated such that they allhave the same sign on their y component) and transform it to a camerapan angle θ_(p) such that the average displacement is aligned with thephysical space y axis:

θ_(p)=tan⁻¹(d _(x) /d _(y))

The camera parameters, as derived by the calibration process applied bythe processing logic 130 as described herein, are given by the computedquantities

θ_(t) (tilt angle)

θ_(r) (roll angle)

θ_(p) (pan angle)

h (camera installation height)

F (focal length), if it had not been measured prior to camerainstallation

For simplicity, modules of the calibration logic 170, e.g. thepositional information computation module 163, are shown here as singlemodules, but it will be appreciated that the implementation of thesemodules may vary. For example in a software implementation some or allof the logic of the positional information computation module 163 may bedistributed in different modules (e.g. instantiable classes) forperforming different geometric computations and/or some or all of itslogic may be implemented simply as software code of the calibrationmodule. Thus in some implementations some or all of the computationsshown here may be implemented by lines of code directly where they areused, rather than in a separate module (e.g. software class) called uponby a parent process. This is of course also true of the other modules,although for many reasons (e.g. programming efficiency for softwareembodiments, and chip real-estate conservation in hardware embodiments)isolation of the more complicated repeated functions into modules may beconsidered preferable in many implementations..

Although various embodiments have been illustrated, this was for thepurpose of describing, but not limiting, the present invention. Variouspossible modifications and different configurations will become apparentto those skilled in the art and are within the scope of the presentinvention, which is defined more particularly by the attached claims.

1. A method for calibrating a vehicle speed estimation system thatemploys a camera having a set of camera parameters to estimate the speedof vehicles based on images received from the camera, the images definedin an image space and the vehicles and camera each belonging to aphysical space and having an absolute position therein, the methodcomprising: a) receiving at a controller from the camera a calibrationimage stream comprising a first image of a vehicle captured at a firsttime and a second image of the vehicle captured at a second time, thefirst and second images having the same image space; b) at a controllerexecuting controller logic to identify in each of the first and secondimage a vehicle feature having a known dimension in the physical space,the identifying comprising determining a placement of the vehiclefeature in the image space at each of the first and second times; c) atthe controller executing controller logic to define on the basis of theplacement of the vehicle feature in the image space at each of the firstand second times an absolute-position-independent relationship betweenan image space placement and a physical space position relative to thecamera; d) at the controller, using controller logic, applying theabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera in aspeed estimation algorithm to determine the speed of a vehicle capturedby the camera.
 2. The method of claim 1, wherein executing controllerlogic to define on the basis of the placement of the vehicle feature inthe image space at each of the first and second times anabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the cameracomprises: assuming a set of physical constraints concerning the vehiclefeature whereby the absolute-position-independent relationship betweenan image space placement and a physical space position relative to thecamera is further defined on the basis of the set of physicalconstrains.
 3. The method of claim 2, wherein the set of physicalconstraints comprises at least one of: a) a known dimension of thefeature; b) that the vehicle feature is perpendicular to a direction oftravel of the vehicle; and c) that the vehicle feature maintains aconstant vertical distance from the camera.
 4. (canceled)
 5. (canceled)6. The method of claim 2, wherein the absolute-position-independentrelationship between an image space placement and a physical spaceposition relative to the camera is a selectedabsolute-position-independent relationship, and wherein the executingcontroller logic to define the selected absolute-position-independentrelationship comprises: a) assuming an assumed set of camera parameters;b) defining a tentative absolute-position-independent relationshipbetween an image space placement and a physical space position relativeto the camera on the basis of the assumed set of camera parameters; c)evaluating the validity of the tentative absolute-position-independentrelationship and upon confirming the validity establishing the tentativeabsolute-position-independent relationship as the selectedabsolute-position-independent relationship.
 7. The method of claim 6,wherein evaluating the validity of the tentativeabsolute-position-independent relationship comprises d. using thetentative absolute-position-independent relationship computing aphysical space location of at least a portion of the vehicle at each ofthe first and second times; e. verifying compliance with the assumed setof physical constraints of the computed space location of the at least aportion of the vehicle at each of the first and second times.
 8. Themethod of claim 7, wherein: a) the set of camera parameters is a firstset of camera parameters from among a plurality of sets of cameraparameters, further comprising repeating steps a-e with each of theplurality of different sets of camera parameters; b) verifyingcompliance with the set of physical constraints comprises using thetentative absolute-position-independent relationship and the set ofphysical constraints to compute a tentative image space placement forthe vehicle feature and computing an error between the tentative imagespace placement for the vehicle feature and the placement of the vehiclefeature in the image space at at least one of the first and secondtimes; and c) verifying compliance with the set of physical constraintscomprises selecting a selected set of camera parameters from among theplurality of sets of camera parameters providing the lowest error. 9.(canceled)
 10. (canceled)
 11. The method of claim 6, wherein the assumedset of camera parameters comprises at least one of a) a tilt angle(θ_(t)); and b) a roll angle (θ_(r)).
 12. (canceled)
 13. The method ofclaim 6, wherein the assumed set of camera parameters comprises tiltangle (θ_(t)) and a roll angle (θ_(r)), and wherein theabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera is afunction of the matrix M where: $M = {\begin{pmatrix}{{- \cos}\; \theta_{r}} & {{- \sin}\; \theta_{r}} & 0 \\{\sin \; \theta_{t}\sin \; \theta_{r}} & {{- \sin}\; \theta_{t}\cos \; \theta_{r}} & {\cos \; \theta_{t}} \\{{- \cos}\; \theta_{t}\sin \; \theta_{r}} & {\cos \; \theta_{t}\cos \; \theta_{r}} & {\sin \; \theta_{t}}\end{pmatrix}.}$
 14. The method of claim 13, wherein theabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera comprisesthe function f_(y)(X, Y, z) for converting from image-space coordinatesto physical-space coordinates where:${f_{y}\left( {X,Y,z} \right)} = {z{\frac{{{M_{21}\left( {X - c_{X}} \right)}p_{x}} + {{M_{22}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{23}F}}{{{M_{31}\left( {X - c_{X}} \right)}p_{x}} + {{M_{32}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{33}F}}.}}$15. The method of claim 13, wherein the absolute-position-independentrelationship between an image space placement and a physical spaceposition relative to the camera comprises the functions f_(x) ⁻¹(x,y,z)and f_(x) ⁻¹(x,y,z) for converting from physical-space coordinates toimage-space coordinates where:${f_{X}^{- 1}\left( {x,y,z} \right)} = {{\frac{F}{p_{x}}\frac{{M_{11}^{- 1}x} + {M_{12}^{- 1}y} + {M_{13}^{- 1}z}}{{M_{31}^{- 1}x} + {M_{32}^{- 1}y} + {M_{33}^{- 1}z}}} + c_{X}}$${f_{Y}^{- 1}\left( {x,y,z} \right)} = {{\frac{F}{p_{y}}\frac{{M_{21}^{- 1}x} + {M_{22}^{- 1}y} + {M_{23}^{- 1}z}}{{M_{31}^{- 1}x} + {M_{32}^{- 1}y} + {M_{33}^{- 1}z}}} + {c_{Y}.}}$16. A camera-based visual speed estimation system for estimating thespeed of vehicles on the basis of video output from of a camera, thecamera-based visual speed estimation system comprising: a) a controllercamera input for receiving from the video output of the cameracomprising at least first and second images of a travelling vehiclecaptured at respective first and second times, the first and secondimages being defined in an image space and the travelling vehicle andcamera each belonging to a physical space and having respective absolutepositions therein; b) a tangible controller in communication with thecamera input and receiving therefrom the video data from the cameracomprising at least the first and second images, the controllercomprising calibration logic executable on the controller for causingthe controller to: i. identify in each of the first and second image avehicle feature having a known dimension in the physical space, theidentifying comprising determining a placement of the vehicle feature inthe image space at each of the first and second times; ii. define on thebasis of the placement of the vehicle feature in the image space at eachof the first and second times an absolute-position-independentrelationship between an image space placement and a physical spaceposition relative to the camera; iii. apply theabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera in aspeed estimation algorithm to derive speed estimation data indicative ofthe speed of a vehicle captured by the camera; c) a controller output incommunication with the controller for receiving from the controller thespeed estimation data and outputting the speed estimation data.
 17. Thecamera-based visual speed estimation system of claim 16, wherein thecalibration logic executable on the controller for causing thecontroller to define on the basis of the placement of the vehiclefeature in the image space at each of the first and second times anabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the cameracomprises: logic for causing the controller to assume a set of physicalconstraints concerning the vehicle feature whereby theabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera isfurther defined on the basis of the set of physical constrains.
 18. Thecamera-based visual speed estimation system of claim 17, wherein the setof physical constraints comprises at least one of: a) a known dimensionof the feature; b) that the vehicle feature is perpendicular to adirection of travel of the vehicle; and c) that the vehicle featuremaintains a constant vertical distance from the camera according to thephysical space.
 19. (canceled)
 20. (canceled)
 21. The camera-basedvisual speed estimation system of claim 17, wherein theabsolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera is aselected absolute-position-independent relationship, and wherein thecalibration logic executable on the controller for causing thecontroller to define the selected absolute-position-independentrelationship comprises logic executable on the controller for causingthe controller to: a) assume an assumed set of camera parameters; b)define a tentative absolute-position-independent relationship between animage space placement and a physical space position relative to thecamera on the basis of the assumed set of camera parameters; c) evaluatethe validity of the tentative absolute-position-independent relationshipand upon confirming the validity establishing the tentativeabsolute-position-independent relationship as the selectedabsolute-position-independent relationship.
 22. The camera-based visualspeed estimation system of claim 21, wherein the logic executable on thecontroller for causing the controller to evaluate the validity of thetentative absolute-position-independent relationship comprises logicexecutable on the controller for causing the controller to: f. using thetentative absolute-position-independent relationship compute a physicalspace location of at least a portion of the vehicle at each of the firstand second times; g. verify compliance with the assumed set of physicalconstraints of the computed space location of the at least a portion ofthe vehicle at each of the first and second times.
 23. The camera-basedvisual speed estimation system of claim 22, wherein: a) the set ofcamera parameters is a first set of camera parameters from among aplurality of sets of camera parameters, wherein the calibration logicfurther comprises logic executable on the controller for causing thecontroller to repeat steps a-e with each of the plurality of differentsets of camera parameters; b) the logic executable on the controller forcausing the controller to verify compliance with the set of physicalconstraints comprises logic executable on the controller for causing thecontroller to using the tentative absolute-position-independentrelationship and the set of physical constraints to compute a tentativeimage space placement for the vehicle feature and to compute an errorbetween the tentative image space placement for the vehicle feature andthe placement of the vehicle feature in the image space at at least oneof the first and second times; and c) the logic executable on thecontroller for causing the controller to verify compliance with the setof physical constraints comprises logic executable on the controller forcausing the controller to select a selected set of camera parametersfrom among the plurality of sets of camera parameters providing thelowest error.
 24. (canceled)
 25. (canceled)
 26. The camera-based visualspeed estimation system of claim 21, wherein the assumed set of cameraparameters comprises at least one of a) a tilt angle (θ_(t)); and b) aroll angle (θ_(r)).
 27. (canceled)
 28. The camera-based visual speedestimation system of claim 21, wherein the assumed set of cameraparameters comprises a tilt angle (θ_(t)) and a roll angle (θ_(r)), andwherein the absolute-position-independent relationship between an imagespace placement and a physical space position relative to the camera isa function of the matrix Mwhere: $M = {\begin{pmatrix}{{- \cos}\; \theta_{r}} & {{- \sin}\; \theta_{r}} & 0 \\{\sin \; \theta_{t}\sin \; \theta_{r}} & {{- \sin}\; \theta_{t}\cos \; \theta_{r}} & {\cos \; \theta_{t}} \\{{- \cos}\; \theta_{t}\sin \; \theta_{r}} & {\cos \; \theta_{t}\cos \; \theta_{r}} & {\sin \; \theta_{t}}\end{pmatrix}.}$
 29. The camera-based visual speed estimation system ofclaim 27, wherein the absolute-position-independent relationship betweenan image space placement and a physical space position relative to thecamera comprises the function f_(y)(X, Y,z) for converting fromimage-space coordinates to physical-space coordinates where:${f_{y}\left( {X,Y,z} \right)} = {z{\frac{{{M_{21}\left( {X - c_{X}} \right)}p_{x}} + {{M_{22}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{23}F}}{{{M_{31}\left( {X - c_{X}} \right)}p_{x}} + {{M_{32}\left( {Y - c_{Y}} \right)}p_{y}} + {M_{33}F}}.}}$30. The camera-based visual speed estimation system of claim 28, whereinthe absolute-position-independent relationship between an image spaceplacement and a physical space position relative to the camera comprisesthe functions f_(x) ⁻¹(x,y,z) and f_(x) ⁻¹(x,y,z) for converting fromphysical-space coordinates to image-space coordinates where:${f_{X}^{- 1}\left( {x,y,z} \right)} = {{\frac{F}{p_{x}}\frac{{M_{11}^{- 1}x} + {M_{12}^{- 1}y} + {M_{13}^{- 1}z}}{{M_{31}^{- 1}x} + {M_{32}^{- 1}y} + {M_{33}^{- 1}z}}} + c_{X}}$${f_{Y}^{- 1}\left( {x,y,z} \right)} = {{\frac{F}{p_{y}}\frac{{M_{21}^{- 1}x} + {M_{22}^{- 1}y} + {M_{23}^{- 1}z}}{{M_{31}^{- 1}x} + {M_{32}^{- 1}y} + {M_{33}^{- 1}z}}} + {c_{Y}.}}$